If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is
added to it, then the whole straight line has been cut in extreme and mean ratio, and the original
straight line is the greater segment.

If in an equilateral and equiangular pentagon straight lines
subtend two angles are taken in order, then they cut one another in extreme and mean ratio,
and their greater segments equal the side of the pentagon.

If the side of the hexagon and that of the decagoninscribed in the same circle are added together,
then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to
prove that the square on the side of the icosahedron is the irrational straight line called minor.

Corollary
The square on the diameter of the sphere is five times the
square on the radius of the circle from which the icosahedron has been described, and the diameter of the
sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle.